Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30871 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2675 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3068 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1377 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6)(7,18)(8,22)(9,15)(10,28)(11,16)(12,17)(13,20)(14,26)(19,32)(21,30)(23,35)(24,31)(25,34)(27,33)(29,36);(5,7)(6,13)(8,23)(9,10)(11,20)(12,21)(14,31)(15,17)(16,30)(18,28)(19,25)(22,29)(24,27)(26,36)(32,33)(34,35);(5,8)(6,14)(7,19)(9,24)(10,25)(11,26)(12,27)(13,29)(15,32)(16,33)(17,34)(18,35)(20,23)(21,36)(22,28)(30,31);(5,9,11,21,12,20,10,7)(6,15,28,18,17,13,16,30)(8,24,26,36,27,23,25,19)(14,32,22,35,34,29,33,31)Polycyclic presentation:
2^9|a^2=h,d^2=ei,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,f^c=fi,g^c=gi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290754);